Embedded newform 731.2.s.b.214.15

• Label: 731.2.s.b.214.15
• Level: $731$
• Weight: $2$
• Character: 731.214
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $496$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.s (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(496\)
Relative dimension:	\(62\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

Embedding invariants

Embedding label			214.15
Character	\(\chi\)	\(=\)	731.214
Dual form			731.2.s.b.386.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.44571 - 0.598831i) q^{2} +(-0.336754 + 0.503988i) q^{3} +(0.317255 + 0.317255i) q^{4} +(0.505403 + 2.54083i) q^{5} +(0.788652 - 0.526960i) q^{6} +(0.214808 - 1.07991i) q^{7} +(0.928987 + 2.24277i) q^{8} +(1.00745 + 2.43220i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 496 q - 16 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/731\mathbb{Z}\right)^\times\).

\(n\)	\(173\)	\(562\)
\(\chi(n)\)	\(e\left(\frac{3}{16}\right)\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.44571	−	0.598831i	−1.02227	−	0.423438i	−0.192353	−	0.981326i	\(-0.561612\pi\)
							−0.829916	+	0.557888i	\(0.811612\pi\)
\(3\)	−0.336754	+	0.503988i	−0.194425	+	0.290978i	−0.915855	−	0.401510i	\(-0.868485\pi\)
							0.721429	+	0.692488i	\(0.243485\pi\)
\(5\)	0.505403	+	2.54083i	0.226023	+	1.13629i	0.912479	+	0.409123i	\(0.134165\pi\)
							−0.686456	+	0.727171i	\(0.740835\pi\)
\(7\)	0.214808	−	1.07991i	0.0811899	−	0.408169i	−0.918722	−	0.394905i	\(-0.870778\pi\)
							0.999912	−	0.0132646i	\(-0.00422237\pi\)
\(11\)	−3.15046	−	4.71500i	−0.949900	−	1.42163i	−0.906328	−	0.422575i	\(-0.861126\pi\)
							−0.0435722	−	0.999050i	\(-0.513874\pi\)
\(13\)	3.71013	−	3.71013i	1.02900	−	1.02900i	0.0294371	−	0.999567i	\(-0.490629\pi\)
							0.999567	−	0.0294371i	\(-0.00937148\pi\)
\(17\)	0.0834311	+	4.12226i	0.0202350	+	0.999795i
							\(19\)	6.59873	+	2.73328i	1.51385	+	0.627058i	0.976348	−	0.216203i	\(-0.0693673\pi\)
0.537504	+	0.843261i	\(0.319367\pi\)
\(23\)	−2.34156	+	1.56458i	−0.488248	+	0.326237i	−0.775208	−	0.631705i	\(-0.782355\pi\)
							0.286960	+	0.957942i	\(0.407355\pi\)
\(29\)	−6.67159	+	1.32706i	−1.23888	+	0.246429i	−0.770698	−	0.637201i	\(-0.780092\pi\)
							−0.468186	+	0.883630i	\(0.655092\pi\)
\(31\)	−0.767008	+	1.14791i	−0.137759	+	0.206170i	−0.893934	−	0.448199i	\(-0.852066\pi\)
							0.756175	+	0.654369i	\(0.227066\pi\)
\(37\)	−6.24892	+	9.35217i	−1.02732	+	1.53749i	−0.196838	+	0.980436i	\(0.563067\pi\)
							−0.830478	+	0.557052i	\(0.811933\pi\)
\(41\)	7.98673	+	1.58866i	1.24732	+	0.248107i	0.774228	−	0.632907i	\(-0.218138\pi\)
							0.473090	+	0.881014i	\(0.343138\pi\)
\(43\)	6.55003	+	0.311714i	0.998870	+	0.0475360i
							\(47\)	6.05997	−	6.05997i	0.883938	−	0.883938i	−0.109994	−	0.993932i	\(-0.535083\pi\)
0.993932	+	0.109994i	\(0.0350832\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	731.2.s.b.214.15	✓	496
17.12	odd	16	inner	731.2.s.b.386.48	yes	496
43.42	odd	2	inner	731.2.s.b.214.48	yes	496
731.386	even	16	inner	731.2.s.b.386.15	yes	496

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.s.b.214.15	✓	496	1.1	even	1	trivial
731.2.s.b.214.48	yes	496	43.42	odd	2	inner
731.2.s.b.386.15	yes	496	731.386	even	16	inner
731.2.s.b.386.48	yes	496	17.12	odd	16	inner